Sarkozy's theorem in function fields

@article{Green2016SarkozysTI,
  title={Sarkozy's theorem in function fields},
  author={B. Green},
  journal={arXiv: Number Theory},
  year={2016}
}
  • B. Green
  • Published 2016
  • Mathematics
  • arXiv: Number Theory
S\'ark\"ozy proved that dense sets of integers contain two elements differing by a $k$th power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of S\'ark\"ozy's theorem for polynomials over $\mathbb{F}_q$ with polynomial dependencies in the parameters. More precisely, let $P_{q,n}$ be the space of polynomials over $\mathbb{F}_q$ of degree $ 2q^{(1 - c(k,q))n}$, then $A$ contains distinct polynomials $p(T), p'(T)$ such that $p… Expand
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